§ 2.1 Field of dreams
Describe numbers used in common algebra.
The set (or collection) of numbers which algebra uses is called a
"field." You might liken this to a farmer's field of soil from
which corn grows. The mathematician's field is the set of numbers and properties from
which algebra grows.
This field includes all of the numbers we've discussed thus far:
the whole numbers and their ratios (called "the rationals" or fractions);
it also contains numbers we haven't discussed yet: the irrationals (the
non-fractions such as "the square root of two" discussed later, or pi,
3.1415... discussed in Geometry: Perimeter, Area, Volume). But the field of numbers
also contains all of these numbers' negative counterparts. For example, 5 and -5, ½ and
-½, etc.
Up until now, we have said that we cannot subtract a
larger number from a smaller. In a concrete sense, it seems impossible. I cannot
take 7 pennies from a bag of 5. This is a physical interpretation of subtraction.
Mathematics is more abstract than this. Figuratively speaking, an accountant subtracts
what he can and marks in red ink what he cannot. (Red and black rods were actually used
for signed number calculations by the Chinese as early as 500 B.C.!) So for example, the
difference 2, for 5 - 7, could have been written in red ink to indicate that a larger
number had been subtracted from a smaller. A mathematician, on the other hand, marks the
balance, not in red, but with a negative sign (-). So she writes 5 - 7 = -2. This is read
in words as "five minus seven equals negative two."
Introducing the negative sign into our number system creates new
numbers: namely, the "negatives" and the old numbers we can now call the
"positives."
Exercise 2.1
Compute the following:
| [1.] 5 - 7 = ____ |
|
[2.] 10 - 20 = ____ |
| [3.] 2 - 1 = ____ |
|
[4.] 1 - 2 = ____ |
Answer "yes" or "no."
[5.] Is it true that whenever you take a larger number from a
smaller, you will always get a negative number?
[6.] Does algebra use negative fractions?
Write your answer.
[7.] Give an example of how negative numbers could be used in
everyday life.
[8.] Back in 1489, Johann Widmann used the subtraction sign (-)
to label negative numbers. Why do you suppose he chose to use this symbol?
[9.] Describe how to take a larger number from a smaller.
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